One approach that you might find useful/interesting would be Bayesian VARs.
An example of a Bayesian VAR being used to predict inflation can be found here. In this paper, Tim Cogley, Sergui Morozov, and Tom Sargent motivate some methodology for Bayesian VARs and then compare their results to what the Bank of England uses in their "fan charts" (which they use to predict inflation). Cogley and Sargent have two other papers on inflation that are related to this paper, "Drifts and Volatilities" and "Evolving Post-World War II U.S. Inflation Dynamics." This series of papers is a good expose on some of the theory of inflation and how to statistically approach it.
I believe @FooBar is right that most banks are using large-scale VARs (similar to the aforementioned) and New Keynesian models. This isn't directly related to the question, but there is a literature on model averaging in which they can combine the forecasts of VARs and N-K models to provide a "more informed" (I use the phrase more informed here lightly but I do think the model averaging is helpful) prediction as to what will happen. An example of such a paper can be found here